We study a generalized version of the Hopfield model [1]} including multiple proofreading steps. A replicative enzyme \(E\)  can bind to free monomers of type \(x\in \lbrace r,w\rbrace \) , where \(r\)  represents a right monomer and \(w\)  a wrong one. The enzyme may then reject the monomer based on its binding free energy \(\Delta E_x\) . In the incorporation pathway, the enzyme sequentially adopts \(N\)  configurations \(m_i^x\) , \(i=1\dots N\)  before the final incorporation of monomer \(x\) , where \(i=1\)  denotes the initial binding, see Figure REF (a).
